The Gibbs Phenomenon in Fourier Analysis, Splines and Approximations and Its Applications

Managing Editor:

M.HAZEWINKEL Centre/or Mathematics and Computer Science, Amsterdam, The Netherlands

Volume 446 The Gibbs Phenomenon in Fourier Analysis, Splines and Wavelet Approximations

by

Abdul J. Jerri Department of Mathematics and Computer Science, Clarkson University, Potsdam, New York, U.S.A.

Springer·Science+Business Media, B.V A C.I.P. Catalogue record for this book is available from the Library of Congress.

ISBN 978-1-4419-4800-7 ISBN 978-1-4757-2847-7 (eBook) DOI 10.1007/978-1-4757-2847-7 Softcover reprint of the hardcover 1st edition 1998 Printed on acid-free paper

All Rights Reserved

© 1998 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1998 . No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner To my caring wife

Suad

with gratitude Contents

Preface xiii

ACKNOWLEDGEMENTS xvii

AIM OF THE BOOK xix

1 INTRODUCTION 1 1.1 The Gibbs-Wilbraham Phenomenon 1 1.2 Some Basic Elements of Fourier Analysis. 3 1.3 Illustrations and Analysis ...... 12 A. The Truncated Fourier Approximation 12 B. The Truncated Fourier Integral Approximation 16 1.4 Filtering via the Fejer Averaging 26 A. The Fejer Averaging ...... 28 B. The (C, a) Summability ... . 31 1.5 The Lanczos-Local-Type Filtering 34

2 ANALYSIS AND FILTERING 37 2.1 The Truncated Fourier Integral ...... 38 2.2 The Fourier Trigonometric Polynomial...... 40 A. A Note Concerning the General Orthogonal Expansion 43 2.3 The Two Basic Methods of Filtering ...... 44 A. The Lanczos-Local-Type u-Averaging ...... 45 B. The Method of Fejer Averaging and Summability 55 2.4 Transform Methods of Filtering ...... 56 A. The Gegenbauer Transform Method for the Truncated ...... 57 B. The. Truncated Fourier Integrals . . 67 2.5 Examples of Other Filters ...... 73 The Fourier Series in Two Dimensions 78 2.6 Some Advantages for Edge Detection. 80

vii viii CONTENTS

2.7 A Historical Note ...... 83 2.8 The Higher Dimensional Case. .100

3 THE GENERAL ORTHOGONAL EXPANSIONS 107 3.1 A Brief Overview...... 107 3.2 Orthogonal Series Expansions ...... 109 A. The Sturm-Liouville Problem ...... 110 B. The Fourier-Jn-Bessel Series Expansion. . 112 C. The Hankel Transform of Radially Symmetric Func- tions in n Dimensions ...... 119 D. The Classical Orthogonal Polynomials Expansion . 122 E. The Legendre Polynomials Series . . 122 F. The Tchebychev Polynomials Series . 125 G. The Laguerre Polynomials Series . . 127 H. The Hermite Polynomials Series . . . 130 3.3 The Asymptotic Relation to Fourier Series . . 131 Rate of Convergence of the Sturm-Liouville Eigenfunc- tions Expansion...... 137 Singular Sturm-Liouville Problem...... 140 3.4 The Global Effect on the Convergence in Rn ...... 148 A. The Laplacian in n-Dimensional-Fourier Series of Ra- dial Functions...... 148 B. The 3-Dimensional Case ...... 150 C. the Fourier Integral Representation in n-Dimensions . 155 3.5 Filtering for Orthogonal Expansions ...... 156 A. The Fejer Averaging ...... 156 B. A Lanczos-Like a-Factor for General Orthogonal Ex- pansions ...... 157 1. A Lanczos-Like a-Factor for Fourier-Jm-Bessel Series . 159 2. Orthogonal Polynomials Expansions . . 171 3. Integral Transforms Representations...... 177

4 SPLINES AND OTHER APPROXIMATIONS 183 4.1 The Piecewise-Linear Approximation. . 184 4.2 High Order Splines Approximation . 191 4.3 Approximation in Lp-Sense . . . 199 4.4 The Interpolation of the DFT . . . . 203

5 THE WAVELET REPRESENTATIONS 207 5.1 and Fourier Analysis ...... 207 A. The Possible Reason Behind the Gibbs Phenomenon . 207 CONTENTS ix

B. Illustration of Some Basic Wavelets, their Fourier Trans­ forms and a Glimpse at the Gibbs Phenomenon .. 216 5.2 Elements of Wavelet Analysis ...... 222 A. The Continuous Wavelet (Double Integral) Represen­ tation of Functions ...... 222 B. The Discrete Wavelet (Double) Series Expansion of Functions ...... 227 5.3 The Discrete Wavelet Series Approximation ...... 230 A. Preliminaries for Having Discrete Orthonormal Wavelets231 5.4 The Continuous Wavelet Representation ...... 246 A. Detailed Analysis of the Gibbs Phenomenon for Even Wavelets - The Mexican Hat Wavelet ...... 247 B. The Mexican Hat Wavelet and its Gibbs Phenomenon 266 C. Hardy-Functions Wavelets . . 274 D. Recent Preliminary Results ...... 285

REFERENCES 287

Appendix A 297

INDEX OF NOTATIONS 319

SUBJECT INDEX 327

AUTHOR INDEX 335 List of Figures

1.1. The function, period 27r. (Many of the Fig- ures and their basic analysis here are from Jerri [11], Inte- gral and Discrete Transforms with Applications and Error Analysis, 1992. Courtesy of Marcel Dekker Inc...... 13 1.2. The Gibbs phenomenon of the Fourier series partial sum SN(X) of the square wave. N = 10 ...... 13 1.3. A sawtooth function of (1.35) with its clear jump discon­ tinuities. (This and Figs. 1.4, 1.5, 1.7-1.11, and Figs. 2.2-2.4 are from Jerri [11]. Courtesy of Marcel Dekker Inc.) 15 1.4. The initial appearance of the Gibbs phenomenon for the Fourier series approximation of the sawtooth function in (1.35) near the jump discontinuity at x = 2. N = 1,3,5. . 15 1.5. The signum function sgn(t), -00 < t < 00...... 16 1.6. The Gibbs phenomenon of the Fourier integral approxi­ mation sgnB(t) of the signum function sgn(t). B = 40 ... 17 1.7. The sine integral Si(t) and the essence of the Gibbs phe- nomenon near the jump discontinuity at t = O...... 18 1.8. Basic functions with jump discontinuities in the interior of their domains. (a) sgn(t), (b) u(t), (c) Pa(t)...... 20 1.9. A function with a jump discontinuity expressed with the aid of the shifted unit step function. (a) f(t) = 9c(t) + Ju(t - to). (b) 9c(t). (c) u(t - t~)...... 21 1.10. Truncation by the gate function window. The windowing effect: Gb(J) approximation of G(J)...... 22 1.11. The effect of increasing the width of the window for a continuous input signal. (a) Small window width, (b) Large window width...... 24 1.12. The Gibbs phenomenon of sn(x) in (1.57) of approximat- ingthe square wave (see (1.34)) and the Fejer <;l.veraging Sn(x) of (1.63) (or (1.66) for reducing it. n = 40...... 31

xi xii LIST OF FIGURES

1.5. The signum function sgn(t), -00 < t < 00...... 38 1.6 The Gibbs phenomenon of the Fourier integral approxima- tion of sgnB(t) of the signum function sgnB(t). B = 20. . 39 1.7. The Sine integral Si(t)...... 39 2.1. The square wave, its Fourier series approximation and the Gibbs phenomenon, N = 10 (the same as Fig. 1.2) .... , 40 2.2. The Gibbs phenomenon of the square wave. N = 10. (This and Figs. 2.3, 2.4 are from Jerri [11]. Courtesy of Marcel Dekker Inc.) ...... 42 2.3. A possible remedy for the Gibbs phenomenon-approximating sgn(t) by a sgnul (t)...... 45 2.4a. The Gibbs phenomenon of the square wave with its (con­ tinuous) 0'1 (averaging) remedy. N = 10...... 52 2.4b. The square wave with the 0'1 averaging of the Gibbs phenomenon. N = 22...... 52 2.4c. The Gibbs phenomenon and its remedies of one (ad and two (0'2) averagings, N = 40...... 53 2.4d. The 0'1,0'3 and 0'6 averagings of the Fourier series ap- proximation SN(t) of the square wave in (2.3)...... 54 2.5. A comparison between the Fejer averaging and the Lanczos- local-0'1 averaging of the Gibbs phenomenon. n = 20. 56 2.6. The Gibbs phenomenon of the truncated Fourier series (2.21) of the sawtooth function, and its absence for the Gegenbauer expansion of (2.33). (Reprinted from [42] with kind permission from Elsevier Science. NL Sara Burgerharstraat 25, 1055 KV Amsterdam, The Nether­ lands.) ...... 62 2.7. The sawtooth function of (2.98). (Most of the figures here are from Hewitt and Hewitt [lO,Fig. 2]. Courtesy of Archives for History of Exact Sciences - Springer-Verlag Inc.) ...... 85 2.8. The sawtooth-like function of (2.99) (from Hewitt and Hewitt [10, Fig. 1.]. Courtesy of Archives for History of Exact Sciences - Springer-Verlag Inc.)...... 85 2.9. The square wave function of (2.100) (From Hewitt and Hewitt [10, Fig. 3]. Courtesy of Archives for History of Exact Sciences - Springer-Verlag Inc.) ...... 86 LIST OF FIGURES xiii

2.10. The different behavior of the undershoots (minima) near x = 0 for SN(X) of (2.99). (From Fig. 10 in Hewitt and Hewitt [10, p. 145]. (Courtesy of Archives for History of Exact Sciences - Springer-Verlag Inc.) ...... 87 2.11a,b. Two cases of Wilbrahm's (correct)11illustrations of the overshoots and undershoots. (Figs. 2.11 and 2.12 are from [10]). Courtesy of Archives for History of Exact Sciences - Springer-Verlag Inc.) ...... 88 2.12a,b. Hewitt and Hewitt's (modern) recomputations ofWilbra­ ham's two illustrations in Fig. 2. 11a,b. (From [10]. Cour- tesy of Archives for History of Exact Sciences - Springer­ Verlag Inc.) ...... 89 2.13. Graphs of the harmonic analyzer of Michelson and Strat­ ton's [4] for SN(X) of the square wave in (2.100) for N = 1,3,5,7,21 and 79. Note the two cases of N = 79,21 for the Gibbs phenomenon. (From Hewitt and Hewitt [10, Fig. 16]. Courtesy of Archives for History of Exact Sciences - Springer-Verlag Inc.)...... 95 2.14. Hewitt and Hewitt's numerical computations of SN(X) of (2.100) for N = 1,3,5,7,21 and 79 to be compared with Fig. 2.13 of the drawings (graphs) of Michelson and Stratton's harmonic analyzer. (From Hewitt and Hewitt [10, Fig. 18]. Courtesy of Archives for History of Exact Sciences - Springer-Verlag Inc.)...... 96

3.1a. The Gibbs phenomenon at x = 1/2 of the truncated Fourier-Jo-Bessel series (3.28). '"1 = 0.5, N = 20. (This and Figures 3.2-3.3 are from Jerri [12]. Courtesy of Springer­ Verlag Inc.) ...... 115 3.1b: The application of the new O"-factor with its repetitions 0"2 = 0"2, 0"3 = 0"3, 0"5 and their comparison with the Fejer averaging of SN(X) in (3.28). '"1 = 1, N = 50. (The same as Figure 3.15.) ...... 116 3.2. The Gibbs phenomenon of the Fourier-Jo-Bessel series of the square wave function in (3.27) on (0,1), and its "repetition-like" via (3.28) on (0,4). '"1 = 0.5, N = 20. (From Jerri [12,22]. Courtesy of Springer-Verlag Inc. for [12], and the AMS for [22].) ...... 117 xiv LIST OF FIGURES

3.3. The Disappearance of the undershoots (overshoots) at x = 0 of the truncated Fourier-Jo-Bessel series (3.28) as N becomes large. N = 100. (From Jerri [12,22]. Courtesy of Springer-Verlag Inc. for [12], and the AMS for [22].) .. 118 3.4. A (translated) gate function and its truncated Jo-Hankel transform Pa,B(X) of (3.39) with its Gibbs phenomenon around the jump discontinuity at x = 1. B = 50. (From Jerri [22]. Courtesy of the AMS.) ...... 121 3.5. The Gibbs phenomenon in the partial sum SN(X) of the Fourier-Legendre expansion (3.46) of (the square wave) f(x) in (3.45). N = 10...... 123 3.6. The effect of the jump discontinuity at the interior point x = 0 on the convergence of the Fourier-Legendre series at the end points x = ±1. N = 10,40...... 124 3.7. The Gibbs phenomenon in the partial sum SN(X) of the Tchebychev series in (3.53) of (the square wave) in (3.45). N = 10,40...... 126 3.8a. The SN(X) Laguerre polynomials partial sum in (3.60) for approximating f(x) in (3.59). N = 30,60,80...... 128 3.8b. The SN(X) partial sum of the Laguerre polynomial ex­ pansion of the Heaviside function H(x - 2) with its jump discontinuity at x = 2. N = 30, 70...... 129 3.9. The SN(X) Hermite partial sum (3.66) for approximating sgn(x), -00 < x < 00. N = 50 ...... 131 3.10. The (oscillatory) divergence of the three dimensional (radial) Fourier series (3.129) of f(r) = 1, 0 < r < 1, at r = o. N = 20. (From Pinsky [27]. Courtesy of the AMS.) ...... 152 2.3. A possible remedy for the Gibbs phenomenon via approx­ imating sgn(t) by a continuous function sgnul (t) of (2.12). 158 3.11. Reducing the Gibbs phenomenon in SN(X) of (3.148) via approximating the function f(x) of (3.149) by the continuous one g(x) of (3.150). N = 5. (This and most of Figures 3.12 - 3.30 are from Jerri [41]. To be submitted.) 161 3.12. The application of the u-factor to the Fourier-Jo-Bessel series (3.153). N = 50 ...... 163 3.13. The repeated application of the u-factor. u, u 2 , u 3 and u5 for the Fourier-Jo-Bessel series in (3.153). N = 50. .. 163 3.14. The u, u 2 and u 3 factors for reducing the Gibbs phe­ nomenon in the Fourier-Jo-Bessel series (3.148). N = 50 .. 164 LIST OF FIGURES xv

3.15. Comparison of filtering the Gibbs phenomenon in Jo­ Bessel series (3.148) via the Fejer averaging (3.145) versus the present q, q2, q3 and q5-factors. N = 50...... 165 3.16. Reducing the Gibbs phenomenon via approximating the square wave-like function f(x) of (3.148) by h(x) of (3.157). N = 5...... 167 3.17. The Gibbs phenomenon in SN(X) of (3.156a) and the q, q2, q3, and q5-factors applications for reducing it in the Fourier-J1-Bessel series (3.158). N = 20 ...... 168 3.18. The q, q2 and q3-factors for reducing the Gibbs phe­ nomenon in the Fourier-J1-Bessel series (3.156). N = 20 .. 168

3.19. Comparison of filtering the Gibbs phenomenon in J1- Bessel series (3.156a) via the Fejer averaging (3.145) ver- sus the present q, q2, q3, and q5-factors. N = 20. . ... 169 3.20. The Gibbs phenomenon and its reduction via the use of the q-factor in the Fourier-J2-Bessel series (3.161), and its repeated applications q2, q3, and q5. N = 20...... 170 3.21. Comparison of filtering the Gibbs phenomenon in the Fourier-J2-Bessel series (3.161) via the Fejer averaging of (3.145) and the present q,q2, q3 and q5-factors. N = 20. 170 3.22. Reducing the Gibbs phenomenon via approximating the square wave of (3.45) by ')'(x) of (3.164). N = 4...... 172 3.23. The Gibbs phenomenon of the Legendre polynomial ex­ pansion (3.46) and its reduction via the q-factor of (3.166) and its repeated applications of q2, q3, and q5. N = 30. . 173 3.24. The q, q2 and q3-factors for reducing the Gibbs phe­ nomenon in the Legendre polynomials expansion (3.46) of the square wave in (3.45). N = 30 ...... 174 3.25. Comparison of filtering the Gibbs phenomenon in the Legendre polynomial expansion (3.46) via the Fejer aver­ aging of (3.46) and filtering via the present q, q2, q3, and q5-factors averagings. N = 30...... 174 3.26. The Gibbs phenomenon of the Tchebychev polynomi- als expansion of f(x) in (3.53) and its reduction via the q, q2, q3 and q5-factors. N = 30...... 176 3.27. Comparison of filtering the Gibbs phenomenon, in the Tchebychev polynomial expansion of the square wave, via the Fejer averaging and the present q, q2, q3, and q5_ factors method. N = 30...... 176 XVI LIST OF FIGURES

3.4. The (translated) gate function Pa(x) and its truncated Jo-Hankel transform Pa,B(X) of (3.39) with its Gibbs phe­ nomenon around the jump discontinuity at x = a = 1. B = 50 ...... 177 3.28. Reducing the Gibbs phenomenon via approximating the (translated) gate function Pl(X) of (3.37), (3.39) by g(x) of (3.173)...... 178

4.1. The square wave of (4.1) and its periodic extension. . 186 4.2. The Sawtooth-like function of (4.40) ...... 199 4.3. The Gibbs variation with P of Lp-sense approx- imation. P = 1,1.5,2.0,4.0,20. n = 20 (From Fig. 1.1 in Moskona, et al. [34]. Courtesy of J. of Constructive Approximation, Springer-Verlag New York, Inc.) . 200 4.4. f(x) with jump discontinuities at x = ~, ~ + 1r...... 204

5.1. The Mexican hat (basic) wavelet. (This and Figs. 5.5,5.8, 5.9 and 5.11, 5.12 are from Daubechies [98]. Courtesy of SIAM.) ...... 210 5.2a. Translation by b > 0 and dilation by a < 1 (compression- high frequency) of the Mexican hat wavelet ...... 211 5.2b. Translation by b < 0 and dilation by a > 1 (stretching- low frequency) of the Mexican hat wavelet...... 211 5.3. The well localized Mexican hat (basic) wavelet 'ljJ(t) in (5.1) and its w{w) in (5.3)...... 212 5.4. The 'ljJ(t) ...... 215 5.5. The Battle-Lemarie scaling function

5.12. The real and imaginary parts of the Poisson wavelet .,pP,2{t) of (5.14). {From Rasmussen [37]. Courtesy of Oxford University Press (Clarendon Press.)) ...... 223 5.13. The Fourier transform 'lIp,2{W) of the Poisson wavelet .,pP,2(t)...... 224 5.14. The scaling function

5.1. The increase ofthe Gibbs first overshoot (percentage) with the order m of the Poisson wavelet used'ljJp,m(t). (From Rasmussen[37]. Courtesy of Oxford University Press (Claren­ don Press).) ...... 285

XIX PREFACE

This book represents the first attempt at a unified picture for the pres­ ence of the Gibbs (or Gibbs-Wilbraham) phenomenon in applications, its analysis and the different methods of filtering it out. The analysis and filtering cover the familiar Gibbs phenomenon in Fourier series and integral representations of functions with jump discontinuities. In ad­ dition it will include other representations, such as general orthogonal series expansions, general integral transforms, splines approximation, and continuous as well as discrete wavelet approximations. The mate­ rial in this book is presented in a manner accessible to upperclassmen and graduate students in science and engineering, as well as researchers who may face the Gibbs phenomenon in the varied applications that in­ volve the Fourier and the other approximations of functions with jump discontinuities. Those with more advanced backgrounds in analysis will find basic material, results, and motivations from which they can begin to develop deeper and more general results. We must emphasize that the aim of this book (the first on the sUbject): to satisfy such a diverse audience, is quite difficult. In particular, our detailed derivations and their illustrations for an introductory book may very well sound repeti­ tive to the experts in the field who are expecting a research monograph. To answer the concern of the researchers, we can only hope that this book will prove helpful as a basic reference for their research papers. In addition, there is always the possibility of following it by a research monograph. To accommodate all those concerned with emphasis on the clarity with some intuition, we shall quote only the very basic theorems, such as those of Fourier and wavelet analysis. The basic and most likely familiar results and theorems of Fourier analysis are reviewed in Chapter 1. We will rely on a good number of the basic references that date back to Wilbraham in 1848. For completeness, we are also including most other references that deal with the Gibbs phenomenon in some way or another. They are placed separately as "Other Related References" in an Appendix following the main bibliography of this book. To distin-

xxi xxii Preface

guish these references from the ones used in the text, we have added a prefix A to their (separate) numbers. These references also include some very recent ones, or few "somewhat" relevant ones, that were discovered too late to be included in our general discussion. For completeness, we shall list such references at the end of their corresponding sections. Aside from a number of additions, including theses done in the US since 1930, and other updatings since 1980, the detailed historical notes will have the spirit of the historical account of E. Hewitt and R. Hewitt, that appeared in 1980. A summary, of this historical account, for the Fourier as well as other representations, is covered in Section 2.7 of Chapter 2. The first chapter starts with a short historical overview of the Gibbs phenomenon, then concentrates on a brief review of the very basic ele­ ments and notations of Fourier analysis. This is followed by a few typical illustrations for the appearance of the Gibbs phenomenon in the trun­ cated Fourier series and integrals. It, then, looks back at Fejer averaging and the essentials of the summability theory, which is of importance to one of the basic methods of filtering the Gibbs phenomenon. This fil­ tering method is complemented by the other very basic method, namely the Lanczos-Iocal filtering. Chapter 2 covers the basic detailed analysis of the Gibbs phenomenon in Fourier series and integrals representations of signals, and the basic methods of filtering it out. It also includes other typical methods of filtering, a recent transform method, and some possible advantage of the presence of the Gibbs phenomenon for edge de­ tection purposes, such as determining the locations of shocks, or sharper edges for the magnetic resonance imaging (MRI) of the defective parts of the heart, for example. In addition, we have included a close to complete but brief historical account of the Gibbs phenomenon in Fourier analysis and some orthogonal expansions of functions with jump discontinuities, for a time span of over a century. Chapter 3 is devoted, primarily, to general orthogonal series expansion, and the general integral represen­ tation of signals. There is also our most recent attempt at a Lanczos - like-local filtering besides the well known Fejer averaging (or Cesaro) summability methods. Such orthogonal expansions include the Fourier­ Bessel series expansion, and the typical orthogonal polynomials series expansions such as the Legendre, the Tchebychev, the Hermite, and the Laguerre polynomials series expansions. The general integral trans­ forms representation is illustrated with the Hankel transform. Chapter 4 is a short one that starts with the piecewise-linear approximation, then moves to its generalization of high order splines approximation of func- xxiii

tions with jump discontinuities. The Fourier series approximations are looked at in light of the usual convergence in the mean (L2 )-sense, as well as the Lp-sense, where the measure of the Gibbs phenomenon de­ pends on p in Lp. The chapter concludes with a rather new topic of the approximation in interpolating the Discrete Fourier Transform (DFT). Chapter 5 deals with the newest topic, for the presence of a Gibbs-like phenomenon in the continuous as well as the discrete wavelets repre­ sentations of signals. The overall result is that for most wavelets, the overshoots and undershoots are much fewer, and are smaller in magni­ tude than the typical ones of Fourier analysis. The emphasis here is on the clarity of an accessible presentation of this new and very important subject. This allows the reader an intuition for the reason behind expect­ ing such a Gibbs phenomenon. We attempt to compare the new bases of wavelets to the traditional trigonometric bases of Fourier analysis. As in the case of the Fourier integral representation, with its relatively simple computations compared to the Fourier series, we will start the chap­ ter with the continuous wavelet integral representation, followed by the discrete (orthonormal) wavelet series and concluded it by detailed anal­ ysis of the Gibbs phenomenon in (few particular) continuous wavelet representations. The chapter concludes with attempts at filtering the Gibbs phenomenon, which are, primarily, Fejer averaging (or summabil­ ity) methods. However, we will have some remarks regarding possible other filters including our recent Lanczos-like-Iocal filtering method. ACKNOWLEDGEMENTS

The idea of writing this book started with the authors book "Integral and Discrete Transforms with Applications and Error Analysis", Mar­ cel Dekker, 1992, where the Gibbs phenomenon was discussed in some detail, along with other basic errors of Fourier series and integrals' ap­ proximation of signals. In the same year the author presented a general talk on "the Gibbs Phenomenon" at the AMS special session that was organized by Professor M.Z. Nashed in Baltimore, MD in 1992. En­ couragements by other colleagues to pursue such a useful exposition followed. To Professor Nashed lowe many thanks. A year later, I re­ ceived an invitation from Professor Michiel Hazewinkel, the editor of this Kluwer Academic Series on Mathematics and Its Applications, to write on a subject related to harmonic analysis. Then, I chose the topic of this book, and a book on "Linear Difference Equations with Discrete Transform Methods" that appeared in 1996. I am thankful to Professor Hazewinkel. The prefinal draft of this book was sent, for prepublication review to a number of interested colleagues, and to experts in the field including authors of papers of basic new research topics in the book. I am grateful to have received their encouraging comments with some corrections and very constructive suggestions. I would like to thank Professors Paul L. Butzer, Richard W. Hamming, Richard E. Hewitt, Costas Karanikas, M. Zuhair Nashed, Roger S. Pinkham, and Gilbert G. Walter for their general comments and valuable suggestions. Many thanks go to Profes­ sors Gilbert Helemberg, Franklin B. Richards and Edward B. Saff for reviewing parts of the book, especially those related to their research. Professor Mark A. Pinsky was most encouraging from the start of the project, and he reviewed most of the book with detailed comments and suggestions, and lowe him my gratitude. Professor Algirdas Bastys did the most detailed review of the book with concrete suggestions and corrections, and he deserves my sincere appreciation. I would like to thank many of my students and colleagues who either read parts of the

xxv XXVI Acknowledgements

manuscript, or attended my lectures on the subjects; and supplied me with useful suggestions. For the historical flavor and completeness of this book, I have bor­ rowed many figures. The authors and publishers deserve my thanks for graciously granting me their permission. I received help in making many of the figures in the book from Mr. Jiangou Liu, Mr. Steve Alexander and Mr. Joseph Hruska, all of them deserve my thanks. I would like to thank Mrs Cindy Smith for typesetting and finalizing this manuscript with patience and a lot of care. Mr. John Martindale of Kluwer Aca­ demic Publishers showed patience and very good understanding, and was very helpful in facilitating this project, and he deserves my sincere thanks. My special thanks go to my wife Suad and my daughter Huda for being supportive and very patient during the long hours it took me away from them to prepare this book. AIM OF THE BOOK

As the first detailed book dedicated to the subject of the Gibbs (or Gibbs-Wilbraham) phenomenon, we have aspired to make its presenta­ tion accessible to the practitioners in fields where this phenomenon may appear in their use of various function representations. This includes the use of Fourier analysis, as well as orthogonal expansions, splines, and wavelet approximations. The book is also directed towards students in these fields, where it can help the entrance to the field by serving for self study; series of lectures; or a complete one semester new course. With its (almost) complete bibliography and comprehensive treatment, it represents a stepping stone for those who aspire for new, deep and more general results. For this, the book raises a number of questions that can serve to ignite such research interest.

XXVll